Group Theory Abstracts

Groups in which the hypercentral factor group is a $T$ - group

James C. Beidleman

Basic properties of the class mentioned in the title will be presented. Let $X$ be the nonabelian group of order $10$ and $C$ be the cyclic group of order $5$ . Put $G=X\times C$ and note that $G$ is not a $T$ -group but is in the class mentioned in the title. This example will motivate some of our results.

DEPARTMENT OF MATHEMATICS
UNIVERSITY OF KENTUCKY
LEXINGTON, KY 40506-0027
clark at ms.uky.edu

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Note on $\mathcal{PST}$ -groups

Matthew Ragland

$\mathcal T$ -groups are the groups in which normality is transitive. One can generalize the concept of a $\mathcal T$ -group by requiring Sylow-permutability to be transitive. We call $G$ a $\mathcal{PST}$ -group if whenever $H$ permutes with the Sylow subgroups of $K$ and $K$ permutes with the Sylow subgroups of $G$ we have $H$ permutes with the Sylow subgroups of $G$ . Let us say $G$ satisfies the property ${\mathcal C}_p$ if every subgroup of a Sylow $p$ -subgroup $P$ of $G$ is normal in $N_G(P)$ . Also, let us say $G$ satisfies the property ${\mathcal Y}_p$ if for every Sylow $p$ -subgroup $P$ of $G$ we have $H\leq K \leq P$ implies $H$ permutes with the Sylow subgroups of $N_G(K)$ . M. Asaad has shown in a recent paper that $G$ is a solvable $\mathcal T$ -group if and only if $G$ satisfies the property ${\mathcal C}_p$ for all primes $p$ dividing the order of $F^*(G)$ , the generalized Fitting subgroup of $G$ . We wish to discuss a proof for the following theorem:

Theorem 1   $G$ is a solvable $\mathcal{PST}$ -group if and only if $G$ satisfies the property ${\mathcal Y}_p$ for all primes $p$ dividing the order of $F^*(G)$ .

This is joint work with Adolfo Ballester-Bolinches and Ramon Esteban-Romero.

DEPARTMENT OF MATHEMATICS
AUBURN UNIVERSITY MONTGOMERY
3348 WALTON DR.
MONTGOMERY AL 36111
mragland at mail.aum.edu

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Transitive and Persistent Subgroups

Joseph Petrillo

Given subgroup properties $\alpha$ and $\beta$ , a subgroup $U$ of a group $G$ may or may not possess one or both of the following properties:

$\alpha\beta$ -transitivity: Every $\alpha$ -subgroup of $U$ is a $\beta$ -subgroup of $G$ .

$\alpha\beta$ -persistence: Every $\beta$ -subgroup of $G$ in $U$ is an $\alpha$ -subgroup of $U$ .

We will present some elementary results and discuss examples of $\alpha\beta$ -transitive and $\alpha\beta$ -persistent subgroups.

ALFRED UNIVERSITY
SAXON DRIVE
ALFRED, NY 14802
petrillo at alfred.edu

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Extremely primitive groups

Ákos Seress

Joint work with Avinoam Mann and Cheryl Praeger

A finite primitive permutation group $G$ is extremely primitive if a point stabiliser $H$ is primitive on each of its orbits. Cyclic groups of prime order, and doubly primitive groups provide infinite families of examples. Efforts to classify the remaining examples (non-regular, simply primitive) are the subject of the lecture. A result of W. A. Manning from 1927 tells us that the stabiliser H acts faithfully on each of its non-trivial orbits. Thus we have an embedding of a primitive group $H$ (or, rather, several primitive actions of $H$ ) in a larger primitive group $G$ . Specific examples of this embedding problem led to the construction of several of the sporadic simple groups, and indeed sporadic groups provide interesting examples of extremely primitive groups. The classification divides into two cases - almost simple and affine - each of which contributes interesting lists of examples. Our aim is to prove that our lists are complete up to a finite number of exceptions. We have achieved this in the affine case; the specificity of our results depends on the strength of asymptotic bounds on the number of maximal subgroups of almost simple groups.

THE OHIO STATE UNIVERSITY
COLUMBUS, OH 43210
akos at math.ohio-state.edu

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Four-variable associative laws for group commutators

Fernando Guzman

In 1941 F. W. Levi proved that a group is nilpotent of class $2$ if and only if the commutator operation is associative. Recently, Geoghegan and Guzman showed that a group is solvable if and only if the commutator operation eventually satisfies all instances of the generalized associative law. In this talk we discuss solvability and nilpotency of groups whose commutator operation satisfies one of the four-variable instances of the associative law.

DEPARTMENT OF MATHEMATICAL SCIENCES
BINGHAMTON UNIVERSITY
BINGHAMTON, NY. 13902-6000
fer at math.binghamton.edu

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R. Thompson's group $V$ from a dynamical viewpoint

Olga Patricia Salazar-Diaz

An element of Thompson's group $V$ can be defined as an automorphism of a certain algebra or as a self homeomorphism of the Cantor set. Thus, the dynamics of an element of $V$ can be studied. We analize the dynamics and use the analysis to give another solution of the conjugacy problem in $V$ .

DEPARTMENT OF MATHEMATICAL SCIENCES
BINGHAMTON UNIVERSITY
BINGHAMTON, NY. 13902-6000
salazar at math.binghamton.edu

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Jordan Decomposition in Integral Group Rings

Lawrence E. Wilson

The Jordan Decomposition is related to writing the Jordan Normal Form of a matrix as the sum of a diagonal matrix and an upper-triangular matrix. Similarly, if the matrix is invertible, you can write it as a product of a diagonal matrix and a unipotent matrix. One can ask whether for integral matrices these two summands (or products) also have integer entries. This question is well-understood for matrices and has been investigated for elements of integer group rings. My co-authors and I have found the complete answer for 2-groups and made important advances when the group order is divisible by a prime at least 5.

CENTER FOR COMMUNICATIONS RESEARCH
4320 WESTERRA CT
SAN DIEGO, CA 92121
larry at ccrwest.org

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Fusion systems of blocks of group algebras

Radu Stancu

Fix $p$ a prime number. Fusion systems on finite $p$ -groups were introduced by L. Puig and provide an axiomatic framework for studying the $p$ -local structure (also called $p$ -fusion) in finite groups. The p-local structure of a finite group $G$ is given by the conjugation with elements of $G$ between the subgroups of a Sylow $p$ -subgroup of $G$ . This axiomatic view point has been very useful in determining many properties of finite groups and of the $p$ -completion of their classifying spaces as well as in modular representation theory and it underlies the theory of $p$ -local finite groups developed by C. Broto, R. Levi and B. Oliver.

So, the $p$ -local structure of a finite group gives a fusion system. But there are examples of fusion systems that do not occur in this way. We call these examples exotic fusion systems. Let $k$ be an algebraically closed field of characteristic $p$ and $G$ a finite group. An interesting question for fusion systems is whether they can be obtained from the local structure of a block of the group algebra $kG$ . In this talk I present a joint work with Radha Kessar on some methods to reduce this question to the case when $G$ is a central $p'$ -extension of a simple group. As an application of our result, we obtain that the exotic fusion systems discovered by A. Ruiz and A. Viruel do not occur as fusion systems of $p$ -blocks of finite groups.

THE OHIO STATE UNIVERSITY
COLUMBUS, OH 43210
stancu at math.ohio-state.edu

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Products of commutators are not always commutators: Cassidy's example revisited

Luise-Charlotte Kappe

In her 1979 paper, entitled ``Products of commutators are not always commutators: an example", P. J. Cassidy presents a group in which the set of commutators is not equal to the commutator subgroup (Monthly, vol. 86, p. 772). However, a typographical error in Cassidy's paper impacts the verification of the claim. In this talk we give a corrected proof of Cassidy's claim in a slightly more general setting and take another look at her example in context with the function $\lambda(G)$ for a group $G$ .

For a group $G$ the function $\lambda(G)$ denotes the smallest integer $n$ such that every element of the commutator subgroup $G$ is a product of $n$ commutators. The statement that the set of commutators of a group $G$ does not form a subgroup is equivalent to $\lambda(G) > 1$ . For every prime $p$ , Guralnick has constructed a $p$ -group $P$ of order $p^{n+n^2}$ with $\lambda(P) = n$ . For Cassidy's group $C$ we have $\lambda(C) = \infty$ . However for given $n$ and any prime $p$ , the group $C$ has a suitable homomorphic image $G = G(n,\rho)$ such that $\lambda(G) = n$ and $G$ has order $p^{f(n)}$ , where $f(n)$ is a linear function in $n$ .

DEPARTMENT OF MATHEMATICAL SCIENCES
BINGHAMTON UNIVERSITY
BINGHAMTON, NY. 13902-6000
menger at math.binghamton.edu

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For a given prime $p$ , what is the smallest nonabelian group whose order is divisible by $p$ ?

Gabriela A. Mendoza

In this talk we will show that for any prime $p>3$ , $\vert PSL(2,p)\vert$ with order $\frac{p(p^2-1)}{2}$ is the smallest nonabelian simple group whose order is divisible by $p$ . For $p=2$ and $3$ the group in question is $A_5$ , the alternating group on $5$ letters.

This answers a question posed at the Zassenhaus Conference 2005. It also strengthens the following result presented at the conference:

Theorem 2   Let $p$ be a prime and let $n(p)$ denote the order of the smallest group in which the elements of order dividing $p$ do not form a subgroup. Then $n(p)=min(\frac{p(p^2-1)}{2},p(kp+1))$ where $k$ is the smallest integer for which $kp+1$ is a prime power.

In our earlier result $n(p)$ was defined only as $min(s(p),p(kp+1))$ , where $s(p)$ is the order of the smallest simple group in which the elements of order $p$ do not form a subgroup.

DEPARTMENT OF MATHEMATICAL SCIENCES
BINGHAMTON UNIVERSITY
BINGHAMTON, NY. 13902-6000
mendoza at math.binghamton.edu

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Nonsolvable Groups Satisfying the One-Prime Hypothesis (Part 1)

Donald L. White

(Joint work with Mark L. Lewis)

We say that a finite group $G$ satisfies the one-prime hypothesis if the greatest common divisor of every pair of distinct ordinary irreducible character degrees is either $1$ or a prime. In earlier work, Lewis was able to prove a bound on the number of character degrees for a solvable group $G$ satisfying the one-prime hypothesis and to show that the bound is best possible.

The aim of this paper is to classify the nonsolvable groups satisfying the one-prime hypothesis and to bound the number of character degrees for such groups. In the first part, we determine all almost simple groups, i.e., groups $G$ satisfying $S\leq G\leq {\rm Aut}(S)$ for a nonabelian simple group $S$ , satisfying the one-prime hypothesis. In particular, we show that $S$ must be one of $A_7$ , the Suzuki group ${^2B}_2(8)={\rm Sz} (8)$ , or ${\rm PSL}_2 (q)$ with $q \geq 4$ a power of a prime.

DEPARTMENT OF MATHEMATICAL SCIENCES
KENT STATE UNIVERSITY
KENT, OH 44242
white at math.kent.edu

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Nonsolvable Groups Satisfying the One-Prime Hypothesis (Part 2)

Mark L. Lewis

(This is joint work with Don White.)

Let $G$ be a finite group, and let ${\rm cd} (G)$ be the set of irreducible character degrees of $G$ . We say that $G$ satisfies the One-prime hypothesis if whenever $a,b \in {\rm cd} (G)$ and $a \ne b$ , then the greatest common divisor of $a$ and $b$ is either $1$ or a prime. In an earlier series of papers, I showed that if $G$ is solvable and satisfies the one-prime hypothesis, then $\vert{\rm cd} (G)\vert \le 9$ , and I gave examples of solvable groups satisfying the one-prime hypothesis where the bound is met. In this talk, I will show that if $G$ is a nonsolvable group satisfying the one-prime hypothesis, then $\vert{\rm cd} (G)\vert \le 8$ . In particular, I will show how classifying the almost simple groups that satisfy the one-prime hypothesis can be used to classify the nonsolvable groups that satisfy the one-prime hypothesis.

DEPARTMENT OF MATHEMATICAL SCIENCES
KENT STATE UNIVERSITY
KENT, OH 44242
lewis at math.kent.edu

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Bounds on the number of lifts of a Brauer character of a solvable group

James Patrick Cossey

The Fong-Swan Theorem guarantees that every Brauer character of a solvable group $G$ necessarily has a lift (i.e. an ordinary irreducible character whose restriction to $p$ -regular elements is that Brauer character). In this talk we will demonstrate methods for generating many such lifts, and we will show upper and lower bounds on the number of lifts of a Brauer character in terms of the nucleus and vertex of the character, which are certain subgroups of $G$ determined (up to conjugacy) by the Brauer character.

UNIVERSITY OF ARIZONA
6001 E PIMA ST APT 83
cossey at math.arizona.edu

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Orders of elements of subgroups of wreath product $p$ -groups

Jeffrey M. Riedl

A finite group having a unique minimal normal subgroup is said to be monolithic. Given a prime $p$ and positive integers $d$ and $e$ , we seek to classify up to isomorphism all the monolithic subgroups of the regular wreath product group $Z(p^e)\wr Z(p^d)$ . (This has already been done in case $d=1$ .) In order to recognize when two such subgroups are not isomorphic to each other, efficient methods are needed for calculating isomorphism invariants (such as nilpotence class, order of the center) for each in this vast collection of subgroups. For this purpose, we have determined a general yet reasonably practical method for counting the number of elements of each order in each of these subgroups.

UNIVERSITY OF AKRON
250 BUCHTEL COMMON
AKRON, OH 44325-4002
riedl at uakron.edu

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Inductive arguments for the non-coprime $k(GV)$ -problem

Thomas Michael Keller

The non-coprime $k(GV)$ -problem deals with the general question of how the number of conjugacy classes of the semidirect product $GV$ is bounded in terms of the cardinality of $V$ , where $G$ is a finite group and $V$ is a finite faithful $G$ -module. We present some arguments that may be useful in an inductive approach to this problem.

TEXAS STATE UNIVERSITY
601 UNIVERSITY DRIVE
SAN MARCOS, TX 78666
keller at txstate.edu

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Groups Whose Normalizers Form a Lattice

Joseph Patrick Smith

I will examine groups whose norm is a normalizer within the group. I will start be showing that these groups are nilpotent, which reduces the problem to the realm of $p$ -groups. I will proceed to show that any group that contains the norm is also a normalizer in the group.

DEPARTMENT OF MATHEMATICAL SCIENCES
BINGHAMTON UNIVERSITY
BINGHAMTON, NY. 13902-6000
smith at math.binghamton.edu

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The Layer Lemma

David Arden Jackson

We present a useful construction and lemma for creating finitely generated monoids having specified numbers of ends for their right and left Cayley digraphs.

SAINT LOUIS UNIVERSITY
220 NORTH GRAND, RITTER HALL ROOM 233
ST. LOUIS, MO 63103
jacksoda at slu.edu

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The Weyl Group and Coxeter Elements of $U(n,q)$

Julianne Rainbolt

This talk will review and describe the Weyl group of a finite group of Lie type. The standard example when $G=GL(n,q)$ will be discussed as well as the example when $G=U(n,q)$ . Included will be an examination of some relationships between the Weyl groups of $GL(n,q)$ and $U(n,q)$ including their Coxeter elements.

SAINT LOUIS UNIVERSITY
220 NORTH GRAND, RITTER HALL ROOM 233
ST. LOUIS, MO 63103
rainbolt at slu.edu

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On the Sources of Simple Modules in Nilpotent Blocks

Adam Salminen

I will state Puig's conjecture on the finite number of source algebras for blocks of defect $P$ . Then I will show that for a certain class of Nilpotent blocks we can reduce Puig's conjecture to central $p'$ -extensions of simple groups.

OHIO STATE UNIVERSITY
231 W. 18TH AVENUE
COLUMBUS, OH. 43210
salminen at math.ohio-state.edu

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A Generalization of Supersolvability

Tuval Shmuel Foguel

In this talk we consider a generalization of supersolvability called groups of polycyclic breadth $n$ for $n\geq 1$ , we see that a number of well known results for supersolvable groups generalize to groups of polycyclic breadth $n$ . This generalization of supersolvability is especially strong for the groups of polycyclic breadth $2$ .

DEPARTMENT OF MATHEMATICS
AUBURN UNIVERSITY MONTGOMERY
3348 WALTON DR.
MONTGOMERY AL 36111
tfoguel at mail.aum.edu

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Homogeneous Products of Conjugacy Classes

Edith Adan-Bante

Let $G$ be a finite group and $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a\in G$ in $G$ . Set $[a,G]=\{a^{-1} a^g\mid g\in G\}$ . We can check that $a^G=a[a,G]$ and thus $\vert a^G\vert=\vert[a,G]\vert$ .

Let $a^G$ and $b^G$ be conjugacy classes of $G$ and $a^G b^G=\{xy\mid x\in a^G, y\in b^G\}$ be the product of $a^G$ and $b^G$ . We can check that $a^G b^G$ is a $G$ -invariant subset of $G$ , that is $(xy)^g\in a^G b^G$ for any $xy\in a^G b^G$ and any $g\in G$ . Also, we can check that because $a^G b^G$ is a $G$ -invariant subset of $G$ , then $a^G b^G$ is the union of $n$ distinct conjugacy classes $G$ , for some integer $n$ , and the conjugacy class $(ab)^G$ is a subset of $a^G b^G$ . Set $n=\eta(a^G b^G)$ .

Under what circumstances is $a^G b^G$ also a conjugacy class, i.e. $\eta(a^G b^G)=1$ ? In this talk, we will prove the following

Theorem 3   Let $G$ be a finite group, $a^G$ and $b^G$ be conjugacy classes of $G$ . Assume that $\vert a^G\vert=\vert b^G\vert=\vert(ab)^G\vert$ . Then $a^G b^G =(ab)^G$ if and only if $[a,G]=[b,G]=[ab,G]$ is a subgroup of $G$ .

We will use the previous result to prove the following

Theorem 1   Let $G$ be a finite nonabelian simple group and $a^G$ be a conjugacy class of $G$ . Then $a^G a^G=(a^2)^G$ if and only if $a=1_G$ .

We turn then our attention when $\eta(a^G b^G)>1$ . We wonder what kind of relationships exist among $\eta(a^G b^G)$ , $a^G$ , $b^G$ and the structure of the group $G$ .

We consider the situation where, in addition, the group $G$ is a $p$ -group for some prime $p$ , that is the size of the set $G$ is a power of $p$ . We can check then that given any conjugacy class $a^G$ , the size of $a^G$ is a power of $p$ , that is $\vert a^G\vert=p^n$ for some integer $n$ .

Theorem 4  

Let $G$ be a finite $p$ -group, for some prime $p$ , and $a^G$ be a conjugacy classes of $G$ with $\vert a^G\vert=p^n$ , for some integer $n\geq 0$ . Then

$$\eta(a^G (a^{-1})^G)\geq n(p-1)+1.$$

We will mention then other results in products of conjugacy classes and finite $p$ -groups. The talk will be based on my papers ``On nilpotent groups and conjugacy classes", ``Homogeneous products of conjugacy classes" and ``Products of conjugacy classes and finite $p$ -groups". Those papers are available on www.arxiv.org.

UNIVERSITY OF SOUTHERN MISSISSIPPI GULF COAST
730 EAST BEACH BOULEVARD
LONG BEACH MS 39560
Edith.Bante at usm.edu

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Characterizing injectors in finite solvable groups

Arnold David Feldman

(Joint work with Rex Dark, National University of Ireland, Galway).

We present characterizations of the injectors of a finite solvable group that are independent of the original definition of injector using Fitting sets. This can be considered a solution to a problem posed by Doerk and Hawkes in their book Finite Soluble Groups.

FRANKLIN & MARSHALL COLLEGE
LANCASTER, PA 17604-3003
afeldman at fandm.edu

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A Classification of Certain Maximal Subgroups of Symmetric Groups

Bret Jordan Benesh

Problem 12.82 of the Kourovka Notebook asks for all ordered pairs $(n,m)$ such that the symmetric group Sym$(n)$ embeds in Sym$(m)$ as a maximal subgroup. One family of such pairs is obtained when $m = n + 1$ . Kaluznin and Klin and Halberstadt provided an additional infinite family. This talk will present a third infinite family of ordered pairs and show that no other pairs exist.

HARVARD UNIVERSITY
ONE OXFORD STREET
CAMBRIDGE, MA 02128
benesh at math.harvard.edu

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Finding Eigenvectors in the Minimal Polynomial

Charles Holmes

The following result is well known, but little known at least in most linear algebra texts and even by numerical analysts. Result: Let $A$ be a square matrix with minimal polynomial $m(x)$ and eigenvalue $k$ such that $m(x) = (x-k)q(x)$ . Then the non-zero columns of $q(A)$ are eigenvectors of $A$ .

The proof is trivial. The main significance is that this observation often yields the eigenvectors more easily than finding the reduced row echelon form of $A-kI$ . In particular, $k$ may not be known exactly. Let $\mathbf{k}$ be the approximation for $k$ . Then $A-\mathbf{k}I$ invertible, so the eigenvectors can not be found in the usual way. In this situation reasonably good approximations for the eigenvectors may be found.

MIAMI UNIVERSITY
OXFORD, OH 45056
holmescs at muohio.edu

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An Embedding Theorem for Groups Universally Equivalent to Free Nilpotent Groups

Anthony Michael Gaglione

Let $F$ be a finitely generated (f.g.) nonabelian free group. O. Kharlampovich and A. Myasnikov proved that any f.g. group $H$ containing a distinguished copy of $F$ ($H$ is called an $F$ -group in the algebraic geometry over groups) is universally equivalent to $F$ ( i.e., satisfies precisely the same universal sentences as $F$ does in a first order language appropriate for group theory where all the elements of $F$ are taken as constants - called the language of $F$ ) if and only if there is an embedding of $H$ into a Lyndon's free exponential group, $F^{(Z[t])}$ , which is the identity on $F$ (this is called an $F$ -embedding in the algebraic geometry over groups). Alexei Myasnikov then posed the question as to whether or not a similar result holds for f.g. free nilpotent groups with Lyndon's group replaced with Philip Hall's completion with respect to a suitable binomial ring. We (Dennis Spellman and I) answered Alexei's question in the affirmative. This talk will explain our answer.

DEPARTMENT OF MATHEMATICS
U.S. NAVAL ACADEMY
572C HOLLOWAY ROAD
ANNAPOLIS, MD 21402-5002
amg at usna.edu